Any distribution supported on with a mean differing
by say
from the nearest integer must belong to the set
of the previous section for
. Thus
for large enough
the set
is a subset of
for some positive
.
According to Theorem 1
for all in
.
In view of the inequality
and 7 above we find
On the explosion set there is a (random) such that the right
hand of this
inequality is less than
for all
. A Taylor
expansion of the
logarithm thus shows
for all large and all
in
.
Since
almost surely the lemma is
proved.